High-Frequency Induction- Coil. 321 



where 



m= a+bv ; n= c+d« ; 



fj,= —a + 0i; v=-y + 8i. 



The sum of all such integrated terms, reduced to the trigo- 

 nometrical form, if V be of the form 



V=<>- a '(A cos 0t + B sin 0t) + e~Y'(C cos St + D sin Bt), 

 is 



r _[(A0-BD)(* + 7) + (BC + AD) (0 + 8)] cos (£ + o> 

 H^l +[-(AO-B D) (g + 8)-(BC + AD)(a + 7 )]sin(i8 + 3)/ 

 ( ( a + 7 ) + (£ + £) 



-[(AC + BD)(a + 7 ) + (BC-AD)08-8)]cos(£-o> ) 



+ [(AC + BD)(/9-S)-(BC-AD)(a + 7)lsm(£-o>l 



which, taken between the limits and oo , is 

 ( AC - BD) (a + 7) + (BC + AD) (0 + 8) 



(« + 7) 2 +(£ + S) 2 



(AC + BD)(* + 7)4-(BC-AD)(/3-S) 



or, if a and 7 are so small that they can be neglected in com- 

 parison with and 8, 



BC + AD BC-AD 



+ 8 + /3-S ' 



which is ordinarily small in comparison with the principal 

 terms, and can be neglected. 



(c) V is the sum of harmonic and oscillatory terms. The 

 preceding discussion of case b is immediately applicable by 

 putting y— 0. In general also the period of the oscillation is 

 so much less than that of the harmonic terms that 8 is negli- 

 gible in comparison with 0, and our last expression reduces to 



2BO 



— -~- , which is entirely negligible in comparison with the 



principal terms. 



In the case of the potential in the secondary circuit of our 

 apparatus 



^0 



3 A 2 4- TV 2 P 2 i "T) 2 



V 2^7/ _ ^2 + -t> 2 ^2 +JJ 2 



w T hich becomes, neglecting B 2 and D 2 , and noting that A 2 2 = C 2 2 , 

 A/ A K_ A,»(« + y ) 



4K 3 2 U y)~ 4a 7 K 2 2 • 



